May seem like a noob question: really, why can't we divide by 0?

Question

Yes, I know, can't be answered, blah, blah, blah.... but here are a few of my theories. I know, plenty of other questions like this, but before marking this as a duplicate, consider this, my mathematical friends:

1. We know that $ x/x = 1 $.
2. We also know that $ 0/x = 0 $.

Then, considering these to facts, $$ 0/0 $$ could be

1. $ 1 $, because $ x/x = 1 $
2. or $ 0 $, because $ 0/x = 0 $

Then, of course we can consider $ x/0 $, where $ x $ can be any real number. Then $ 0/0 $ can be $ 0 $ or $ 1 $, and then the rest according to theory, is $ \infty $ or $ \text {undefined} $. Then how come we say $ \dfrac {x}{0} = \infty $? Or is this only true being $ x \neq 0 $?

Or is it that just for practical sense, we just say that $ x/0 = \infty $? Is it because it is just because it is not computationally possible? Or is it just because $$ \dfrac {5}{\dfrac {1}{2}} = 5(\dfrac {2}{1}) = 10 $$ and then $$ \dfrac {5}{\dfrac {1}{100}} = 500 $$ and then $$ \dfrac {5}{\dfrac {1}{100000}} = 500000 $$ and so the numbers keep on going to $ \infty $ as we get closer to $ 0 $?

I know there are lots of possible answers but then the theory that $ \dfrac {x}{0} = \infty $ even though $ \dfrac {0}{0} $ could be 1 or 0, that just does not make sense to me. I will appreciate any answers / at lease possible answers, because I understand that this is just a very controversial topic of mathematics.

Answer 1

This is the way I convinced myself :

$$any number \times 0 = 0\; \rightarrow \! \frac{0}{0}=any number$$

I thinks this is why it is unidentified.

Answer 2

You can find a few good illustrations on why we can't divide by $0$ here. I want to note that infinity is not the answer to $x/0$, what one can say however is that $\lim_{x\rightarrow 0^+} \frac{1}{x}= \infty$. Try it out yourself: compute $1/0.5$, then $1/0.3$, $1/0.1$ etc and you'll see that the answers keep increasing. Because you can keep picking a smaller number to divide by, the limit is infinity.

Answer 3

If you want meaningful arithmetic (for $+$ and $\cdot$), then you want it to obey some rules, for example that $a+0=a$ for all numbers $a$ and the distributive property. These simple rules imply that $a\cdot 0=0$ for all numbers $a$. As division is defined as the inverse operation to multiplication it is clear at this point that you can't define what division by 0 is supposed to mean: $\frac b0$ would be the unique $c$ with $c \cdot 0=b$, but for no $b\neq 0$ is there such a $c$...

Answer 4

The binary operator, division, can be thought of as a statement which asks how many times can "b" go into "a", a/b. If b is zero, I.e. nothing, the operator then asks how many times can nothing go into something?; Well, an infinite number of times. If a and b are zero, then how many times can nothing go into nothing, an infinite number as well, however, this case is less defined then simple division by zero. And, infinity is not a number, it's a concept.